Raffles Institution 04 2024 Y3 SUPP WS Quadratic Functions

Page 1 of 5 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2024 YEAR 3 RP MATHEMATICS TOPIC 4: QUADRATIC FUNCTIONS (MATHS 1 & MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: Sec 3 ( ) Date: 1 2022/Y3RP/M2/T1/Q1 (i) Find the coordinates of the turning point of the curve ( )( )1 5 11 12yx x=+− . [2] (ii) Hence sketch the graph of ( )( )1 5 11 12yx x=+− for 31 x−≤ ≤ , showing all the critical points clearly. [3] [Ans: (i) 32, 655 − ] 2 2022/Y3RP/M2/T1/Q2 The equation of a curve is 2 933y ax x a= ++− , where a is a constant and 0a≠ . (a) Show that the line ( )3y xa= + intersects the curve at two distinct points for all real values of a ( 0a≠ ). [5] (b) Find the range of values of a for which the curve lies completely below the line 3 4y=− . [5] [Ans: (b) 12 4a<− ] 3 2021/Y3RP/M2/T1/Q1 Find the range of values of a such that 2 35x ax a++− is always positive. [3] [Ans: 2 10a<< ] 4 2021/Y3RP/M2/T1/Q2 Find the range of values of k for which the line y kx k+= and the curve ( ) 2 2 1 10y kx k x k=+++ do not meet. [4] [Ans: 11 or 93kk<− > ] 5 2021/Y3RP/M2/T1/Q3 Show that ( ) ( ) ( ) 22 2 3 13 0mx mx m− + + +− = has real and distinct roots for all real values of m such that 2m≠ . [4] 6 2021/Y3RP/M2/T1/Q4 (i) The equation of a curve is ( )( )2 13yx x=+− . Find the maximum value of y and the corresponding value of x. [2] [Ans: (i) 49 5Maximum value of when 84yx= = ]

Page 2 of 5 7 2020/Y3RP/M2/T1/Q1 Find the range of values of p for which the curve ( ) ( ) 2 3 41y x p px=−− − has a positive minimum y value. [4] [Ans: 33 44 p−<< ] 8 2020/Y3RP/M2/T1/Q2 Show that the line ( )43 1yx m= +− cuts the curve ( ) 2 21 2yx m x= + −+ at two distinct points for all real values of m . [5] 9 2019/Y3RP/M2/T2/Q2 Find the range of values of m for which the equation 2 45 4x x mx− += + has no real roots. Hence, determine the number of points of intersection between the curve ( ) 2 2yx= − and the line 23yx= + . [5] [Ans: 62 m− < <− ; 2 points of intersection] 10 2019/Y3RP/M2/T2/Q4 Find the range of values of p for which ( ) 268p xp x− + >− for all real values of x.[5] [Ans: p > 8] 11 2018/Y3RP/M2/T1/Q1 Find the range of values of k such that the equation 2 2 4 4 ( 2)x x kx− −= + has no real roots. [4] [Ans: 112 < 22 k− <− ] 12 2018/Y3RP/M2/T1/Q2 Show that 2232k kx x−++ is positive for all real values of x. [3] 13 2018/Y3RP/M2/T1/Q3 Find the range of values of the constant p for which the line 41yx= − intersects the curve 282 1py x px−= + at two distinct points. [4] [Ans: 1 or 8 and 0pp p<> ≠ ] 14 2017/Y3RP/T2/Q1 Find the range of values of k for which the equation 22 10x kx x k+ +− += has real roots. [3] [Ans: 31 or 5kk≤− ≥ ] 15 2017/Y3RP/T2/Q2 (i) Given that 2 8ax x c−+ is always negative, what condition